Questions involving philosophy of mathematics

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Can number 2 be a particular instance of set membership relation? [on hold]

I have found some definitions of numbers in set theory, such as von Neumann's, Zermelo's, Frege-Russell's, and Cantor's (see e.g. this video https://www.youtube.com/watch?v=6UWhPnbZv-o&sns=fb). ...
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2answers
48 views

What is the difference between a counter-intuitive statement and a paradox?

In mathematics and logic, what is the difference between a counter-intuitive statement and a paradox? For example, what differs something like the Banach-Tarski theorem or Gabriel's horn from ...
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3answers
128 views

The obivious “why”-questions [on hold]

Whenever someone is introduced to a mathematical concept but doesn't stuck to it long enough there are always this "why"-questions. Why is $8+2=2+8$ etc. The answer to that is obvious I think for the ...
4
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1answer
76 views

What's the difference between “unprovable” and “undecidable”?

It seems to me that there is a difference between an unprovable sentence, and an undecidable sentence, but sometimes I have the impression that some authors use the terms interchangeably. In my ...
3
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1answer
55 views

Sheaves in Philosophy

I once found a book on google.books. It was about the applications of sheave theory to philosophy or more general to social studies. I don't remember for sure. i just know it was not the book Sheaves ...
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1answer
64 views

What does if-then has to do with not being true?

I'm reading Chihara's: Constructibility and Mathematical Existence. It says: An even more radical view rejects the assumption that mathematics is true—at least in the straightforward way that ...
3
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5answers
159 views

Does taking courses in mathematics give any help for mathematical logic?

I'm undergraduate student of philosophy department and I think I'll major in mathematical logic. For studying mathematical logic, I thought studying math lectures would give help to logic. So I ...
24
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5answers
2k views

Can mathematics get from other sciences what it got from physics?

Throughout history, physics has been an unparalleled source of '' inspiration'' for discovering/inventing mathematical ideas, which is due to its ability to describe the physical world. But can this ...
2
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2answers
128 views

Is mathematics invented or discovered? [closed]

In physics for example, and in science in general, facts are "discovered" in the sense that they arise from observing nature. A particle is discovered if we can measure its existence in nature. A law ...
4
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1answer
106 views

Are there logical arguments against modern $\sf ZFC$ set theory?

As of asking this question, my knowledge of set theory is quite pedestrian. I've read about it in numerous nontechnical papers and even worked through three chapters of Jech - Set Theory, but in terms ...
4
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1answer
104 views

Is the anti-foundation axiom considered constructive?

In the area of theoretical computer science that I am interested in, constructive mathematics is of practical interest because it gives algorithms that can be implemented on a computer. However, ...
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2answers
80 views

philosophy : first axiom of geometry and variable curvature

The very first axiom of geometry can be described as: Two different points lay on one and only one line. And I was wondering are there surfaces where this axiom irrecoverably fails? and I found ...
34
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11answers
3k views

Why not both true and false?

Why can't some mathematical statement (or whatever is the correct term) be both true and false? For example we can prove (e.g. by induction) that $1+2+3+\cdots+n=\frac{n(n+1)}{2}$ for all positive ...
4
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6answers
1k views

What's behind the Banach-Tarski paradox? [closed]

The discovery of the Banach-Tarski paradox was of course a great thing in mathematics but raises the issue of the relation between mathematics and reality. Empirically there are good reasons for faith ...
15
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4answers
1k views

Why do we need to learn Set Theory?

I was planning to write some article for the Mathematics magazine of our college and it occurred to me that it will be a good idea to write about the impact and importance of Set Theory. I plan ...
4
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6answers
424 views

Question on induction technique

When one uses induction (say on $n$) to prove something, does it mean the proof holds for all finite values of $n$ or does it always hold when even $n$ takes $\pm\infty$?
7
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5answers
185 views

Why do we first introduce the open set definition for continuity instead of the neighborhood definition?

After (nearly) completing my course in topology, something weird just stuck out to me which I hadn't considered before. When first discussing continuity, we often use the following definition: Let ...
2
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1answer
37 views

Properties of the simplest object in n-dimension

In my boredom, I was thinking about why the simplest 3d object (i.e. the one with the least faces, sides, vertices) was the tetrahedron. After it made sense to me, I realized some cool stuff which was ...
4
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2answers
79 views

Why do we focus so much in math on functions (as a subclass of relations)?

Why is it that math so focuses on the subclass of relations known as functions? I.e. why is it so useful for us in nearly all branches of mathematics to focus on relations which are left-total and ...
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2answers
58 views

Are there undecidable problems for which a solution has been found?

I mean are there examples of problems that have been proven to be undecidable, in the sense that it would not be possible to devise a deterministic computer program that outputs a solution for an ...
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3answers
38 views

Is ( Set S contains $x$ and only $x$ then does S equals $x$ ) true?

If a set $S$ contains $x$ and only $x$, then does $S$ equal $x$?
4
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1answer
116 views

Goldbach's conjecture can't be proved to be undecidable?

Conjectures concerning natural numbers which could be settled by a counterexample can, as far as I understand, not be proved to be undecidable without being proved not having a counterexample at the ...
4
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6answers
115 views

the purpose of induction

After getting an answer (in a comment) from peter for this question I have a follow up question. If, in all horses are the same color problem for example, we need to use reason, reason which is ...
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3answers
165 views

How much of Mathematics is limited by our writing? [closed]

I'm sorry if this question is too vague or otherwise a stupid question. Suppose the mathematicians in some alien civilisation similar to ours sculpted their Mathematics in three dimensions (or ...
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0answers
49 views

Is there a name for this constant? (0.0100011011…)

It's the simplest number I could think of that contains any finite binary code in its digits: $$\begin{align} c &= 0.0100011011000001010011100101110111...\\ &= ...
5
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0answers
72 views

Apparent Arbitrariness in Mathematics

Something about definitions in mathematics has always interested – confused? - me, I call it “arbitrariness in Mathematics” - it's a bad name, but I don't know a better one. Let me explain: 1st - ...
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1answer
28 views

maximum number of possible rules of a sequence?

Read this http://www.mathsisfun.com/algebra/sequences-finding-rule.html and also http://en.wikipedia.org/wiki/The_Oxford_Murders_%28film%29 where the scene about the murder note left behind contains a ...
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1answer
39 views

A Question Regarding Representing $\mathscr P$($\omega$) as a Digraph and CH

It is well known that one can represent sets as digraphs. What is the proper digraph representation of $\mathscr P$($\omega$)? I ask this because $\mathscr P$($\omega$) is $\Pi_1$ in the Levy ...
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1answer
54 views

Impredicativity and set theory

I have thought about an example in set theory, but I don't know if its legal to do it, maybe someone can help. Let $\emptyset$ be given and let $A$ be a non empty set. Let us create the subset $X = ...
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1answer
959 views

What is the “opposite” of the Axiom of Choice?

One might think that, trivially, the "opposite" of AC is $\neg$AC. However, thinking about it differently, I'm not sure this is intuitively the case. AC says that every set has a choice function. ...
3
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0answers
38 views

Philosophical implications of P vs NP proof?

Wikipedia article on P vs NP says that "a proof either way would have profound implications for ... Philosophy" without providing further details. So I was wondering what could be the philosophical ...
7
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3answers
223 views

Why is the Power Set Operation Inherently Vague?

It is a somewhat common view among mathematicians/philosophers (who have an opinion on the subject) that the power set operation is inherently vague. They go on to say that its inherent vagueness is ...
5
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1answer
140 views

What goes wrong in the following argument that our conception of “set” is inconsistent?

This question might sound facetious, but it is a genuine question which I am very much interested in. I apologize in advance if it is too conceptual or philosophical, but I'm optimistic that I might ...
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1answer
76 views

Can arithmetic truths fix the truth value of the Continuum Hypothesis?

Many logicians and philosophers believe that all sentences expressible in the language of Peano Arithmetic have determinate truth-values, even though no nice formal system can capture all of these ...
3
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1answer
87 views

Is the first-order incompleteness of a theory (like arithmetics, set theory or logic itself) avoidable in a second or higher-order axiomatizations?

Can we avoid the first-order incompleteness of a theory (like arithmetics or set theory) in a second-order theory which contains the previous? How does it depend on the chosen semantics or models? If ...
4
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0answers
75 views

Strange Consequences of Large Cardinals in Probability

Large cardinal axioms are very strong hypothesizes and as any other strong hypothesis they have many strange consequences in mathematics. On the other hand we know that if we bring even the least ...
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1answer
131 views

Is Bell's Notion of “Abstract Set” Flawed?

Consider the following definition of "abstract set" given by John L. Bell (who wrote the book "Set Theory: Boolean-valued Models and Independence Proofs") from his preprint "Abstract and Variable ...
2
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1answer
77 views

Why the dual of some results are true while others are false?

In mathematics, many results have their "dual" versions. In many cases, if a result is true, then its dual is true as well. However, there are some examples while the dual of a true statement is ...
5
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3answers
374 views

Good Sources for Lecture Movies in Set Theory, Logic and Philosophy of Maths

Of course as any other researcher I'm not able to attend any scientific event in my research area. But it is always interesting and useful to watch the lecture movies of these events. I will ...
0
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4answers
118 views

what is $e$ really? what is its meaning? [duplicate]

I don't get it how we came up with $e$ and how can nature use this number so much! that is what I have been told and I only know that $e$ is a specific constant like $\pi$! I understand that $\pi$ ...
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1answer
77 views

Can a proof be too long? [closed]

Suppose that there is an omnipotent oracle saying that there is a proof for Riemann hypothesis but its proof is so long the universum would collapse before mankind will understand the proof. Would ...
5
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0answers
93 views

Are there examples of mathematical problems proven by abduction?

Proof by deduction is a simple principal. For example: All humans are mortal, and Bill is a human; Therefore, Bill is mortal. However, proof by abduction is a bit different. A famous example: ...
2
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2answers
56 views

Undecidability and truth

Are there undecidable problems for which a single truth exists? For example, the question about parallels is not decidable from Euclid axioms. But multiple answers are valid and give different kinds ...
0
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1answer
177 views

Hand-incalculable Problems

Let's define a "hand-incalculable problem" as a mathematical problem that can not be solved by available human calculation power (using only writing materials and utensils) at a specific date and ...
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7answers
2k views

Why do we stop at exponentiation stage in arithmetic of natural numbers?

In natural numbers the unary successor operator $S$ is the most natural function which maps each number to the next one. Furthermore we may consider the binary relation $+$ as an iteration of $S$. ...
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2answers
81 views

How can we explain the discrepancy between $\rightarrow$ (IF-THEN) and $\setminus$ (A-BUT-NOT-B)?

Let $\mathbb{B} = \{0,1\}$ denote the Boolean domain, ordered in the usual way. Then $\mathbb{B}$ is a lattice. It has a join operation $\vee$ that coincides with "OR," a meet operation $\wedge$ that ...
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1answer
18 views

How could we define the existence of an object/element in the Euclidean space?

Let X be an object/element, What does it mean when I say "X is an object in the Euclidean space"? in other words, What differs an existed object from an unexisted one in the Euclidean space?
3
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2answers
74 views

Formal theories dealing with non-commutattive and non-transitive notion of equality

This question is inspired by a philosophical discussion which I don't want to bother you with. As far as I know when we use (or define) the statement "$x$ is equal to $y$" in logic and ordinary ...
0
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1answer
45 views

Points and real intervals [closed]

The sorites paradox goes like this: Start with a heap of sand. Remove a grain of sand and you still have a heap; remove another, and another, and another, and you'll still have a heap. Eventually, ...
9
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1answer
77 views

Can the set of computable numbers be used as a theoretical basis for calculus?

I recall from my Real Analysis course that the rational numbers $\mathbb{Q}$ are not suitable for doing calculus, and I believe the reason was that $\mathbb{Q}$ does not possess the least-upper-bound ...